Suppose we have $N$ balls and we have an experiment that is set up in the following way.
Your goal is to pick a particular ball that is fixed. At each trial you pick $N^{1-c_1}$ balls and replace. We know that for every pair of trials $N^{1-c_2}$ overlap is there and for every $k$-trials $N^{1-c_2}$ overlap is there. We know that $0<c_1<\dots<c_k<1$ holds.
Alternatively we can view this as follows.
At each trial the probability that you pick the ball is $p_1=1/N^{c_1}$. For any pair of trials the probability that you will pick the same ball is $p_2=1/N^{c_2}$. For any $k$-trials the probability that you will pick the same ball is $p_{k}=1/N^{c_k}$. We know that $0<c_1<\dots<c_k<1$ holds.
Can we upper bound probability that you will not pick the ball after $r$ trials by $(1-p_1p_2)^r$? So probability that you will pick the ball is at least $1-(1-p_1p_2)^r$?